Did you know (I'm sure you did at one point) that a regular decagon can be inscribed in a circle? It turns out that if the radius of the circle is divided into mean-extreme ratio, the larger segment is congruent to the side of the inscribed regular decagon.

Then to inscribe a regular pentagon, of course, one just joins alternate vertices of the inscribed regular decagon.

Now we can prove (or would you rather "conjecture") that a regular pentadecagon can be inscribed in a circle, since we have now have ways of constructing arcs of measure 60 and 36 degrees (by inscribing a regular hexagon and a regular decagon in a circle). Their difference, an arc of measure 24 degrees, is the arc cut off by the side of the regular 15-gon. <------------------------------------(this 19th century geometry person is now off to practice the minuet)----------------<